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Claim. The answer to Problem 56 is no. Write $f(N,k)$ for the largest size of a set $A\subseteq\{1,\ldots,N\}$ with no $k+1$ pairwise coprime elements and $E(N,k)$ for the set of integers up to $N$ divisible by one of the first $k$ primes, the example the question offers. Ahlswede and Khachatrian prove (Proposition 1 and Example 1 of the paper; the repository's reading is on the card Ahlswede and Khachatrian 1994) that if $t$ satisfies $p_{t+7}p_{t+8}<p_tp_{t+9}$ and $p_{t+9}<p_t^2$, then for $k=t+3$ and every $N$ with $p_{t+7}p_{t+8}\leq N<p_tp_{t+9}$ one has $f(N,k)>|E(N,k)|$, and that $t=209$ satisfies both inequalities. So for $k=212$ and $N$ in that range, which lies above $p_k$, the multiples of the first $k$ primes are not the largest set without $k+1$ pairwise coprime elements. The authors add, as an expectation and not a theorem, that known results on gaps between consecutive primes (Erdős's and Rankin's) should show that the inequalities (H) hold for infinitely many $t$, giving such exceptions for arbitrarily large $k$. The paper's positive results, Theorems 1 and 2, concern Erdős's general function $f(n,1,s)$, the largest size of a set of integers up to $n$ coprime to the first $s-1$ primes with no two coprime elements: among squarefree numbers the multiples of $p_s$ are extremal for every $s$ and $n$, and in general for every $n$ large in terms of $s$. The problem's own case $k=1$, like $k=2$, the paper calls easy.

What survives. Erdős then asked whether the conjecture holds once $N$ is large in terms of $k$; the authors' 1995 sequel proves that it does, which is the accepted partial claim Ahlswede and Khachatrian 1995, and leaves the answer to the statement above no.

Formalization. The linked Lean file in Boris Alexeev's repository declares itself a formalization of a solution to the problem whose original human proof is this paper; its header says that ChatGPT (OpenAI) explained a proof of the result, not necessarily the original one, that Aristotle (Harmonic) auto-formalized that text, and that the statement comes from the formal-conjectures project with a hand correction of a missing condition, verified under Lean 4.24.0. The file was first committed on 2025-11-25 at the repository's root and moved under src/ in the reorganizations of 2025-11-26 and 2025-11-27; the link is pinned to the last commit that touched the file at its present path. This corpus has not built or audited the file, so no formalized evidence is listed.

Acceptance. The site's curator, T. F. Bloom, marks the problem disproved and credits this paper for the case $k=212$, which the page lists as reviewed. The paper is R. Ahlswede and L. H. Khachatrian, On extremal sets without coprimes, Acta Arith. 66 (1994), no. 1, 89--99, a refereed journal, listed as refereed. The page is dated by the publication year alone: the publisher's record gives 1994 without a month, so the first day of the year stands in for the issue date.